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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2 --Tarski
 Proof--
Date: Sun, 21 Apr 2024 10:26:51 -0500
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On 4/20/2024 10:39 AM, Richard Damon wrote:
> On 4/20/24 11:20 AM, olcott wrote:
>> On 4/20/2024 2:54 AM, Mikko wrote:
>>> On 2024-04-19 18:04:48 +0000, olcott said:
>>>
>>>> When we create a three-valued logic system that has these
>>>> three values: {True, False, Nonsense}
>>>> https://en.wikipedia.org/wiki/Three-valued_logic
>>>
>>> Such three valued logic has the problem that a tautology of the
>>> ordinary propositional logic cannot be trusted to be true. For
>>> example, in ordinary logic A ∨ ¬A is always true. This means that
>>> some ordinary proofs of ordinary theorems are no longer valid and
>>> you need to accept the possibility that a theory that is complete
>>> in ordinary logic is incomplete in your logic.
>>>
>>
>> I only used three-valued logic as a teaching device. Whenever an
>> expression of language has the value of {Nonsense} then it is
>> rejected and not allowed to be used in any logical operations. It
>> is basically invalid input.
>>
> 
> In other words, you admit that you are being inconsistant about what you 
> are saying, because your whole logic system is just inconsistant.
> 


Not at all.
An undecidable sentence of a theory K is a closed wf ℬ of K such that
neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
not-⊢K ¬ℬ. (Mendelson: 2015:208)

The notion of incompleteness and undecidability requires non truth
bearers to be construed as truth bearers.

A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the primary
bearer of truth or falsity. https://en.wikipedia.org/wiki/Proposition

When we quit construing expressions that cannot possibly be true or
false as propositions then incompleteness and undecidability cease to
exist.

On 4/18/2024 8:58 PM, Richard Damon wrote:
 > INCOMPLETENESS is EXACTLY about the inability to prove statements that
 > are true.

Truth_Bearer(F, x) ≡  ∃x ∈ F ((F ⊢ x) ∨ (F ⊢ ¬x))

....14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)

Gödel is essentially saying that expressions that are not propositions
prove that a formal system of propositions has undecidable propositions.

> You don't seem to understand that predicates, DEFINED to be able to work 
> on ALL memebers of the input domain, must IN FACT, work on all members 
> of that domain.
> 
> For a Halt Decider, that means the decider needs to be able to answer 
> about ANY machine given to it as an input, even a machine that uses a 
> copy of the decider and acts contrary to its answer.
> 
> If you are going to work on a different problem, you need to be honest 
> about that and not LIE and say you are working on the Halting Problem.
> 
> And, if you are going to talk about a "Truth Predicate", which is 
> defined to be able to take ANY "statement" and say if it is True or not, 
> with "nonsense" statements (be they self-contradictory statements, or 
> just nonsense) being just not-true.
> 
> ANY statement means any statement, so if we define this predicate as 
> True(F, x) to be true if x is a statement that is true in the field F, 
> then we need to be able to give this predicate the statemet:
> 
> In F de define s as NOT True(F, s)
> 
> 
> If you claim that your logic is ACTUALLY "two-valued" then if True(F,s) 
> returns false, because s is a statement without a truth value, then we 
> have the problem that the definition of s now says that s has the value 
> of NOT false, which is True.
> 
> So, the True predicate was WRONG, as True of a statement that IS true, 
> must be true.
> 
> If True(F,s) is true, then we have that s is not defined as NOT true, 
> which is false, so the True predicate is again WRONG.
> 
> The predicate isn't ALLOWED to say "I reject this input" as that isn't a 
> truth value (since you claimed you are actually useing a two-valued 
> logic) and this predicate is defined to ALWAYS return a truth value.
> 
> So, it seems you have a two-valued logic system with three logical values.
> 
> Which is just A LIE!
> 
> You are just proving you are too stupid to understand what you are 
> talking about as you don't understand the meaning of the words you are 
> using, as you just studied the system by Zero order principles.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer